29.6.6 Discrete tracer budgets

The purpose of this section is to derive the discrete tracer budgets within the free surface ocean model with a time dependent top cell volume.

As for the formulation of the discrete momentum equations in Section 29.6.2, the continuous time budget for the total amount of tracer within a given tracer cell (Figure 29.2) is given by

$$\partial_{t} \, (A \, h^{t} \, T) = -dy \, (F^{x}_{i+1} - F^{x}_{i}) - A \, (F^{z}_{k-1} - F^{z}_{k}),$$ (29.102)

where T is the tracer per unit volume, i.e., a tracer concentration, A is now the horizontal area of the tracer cell, h^t is the thickness of the tracer cell, meridional gradients are omitted for brevity, and the thickness weighted horizontal advective and diffusive fluxes F^x are computed as in Pacanowski and Gnanadesikan (1998) to account for the generally different adjacent cell thicknesses.

F^z_k is the vertical turbulent and advective tracer flux through the cell interface k. The special term F^z_k=0 for the vertical tracer flux at the sea surface $z=\eta$ involves vertical tracer advection, relative to the undulating sea surface, by fresh water, as well as the usual parameterized turbulent flux

 

$$F^{z}_{k=0} = F^{z,turb}_{k=0} - q_w \, T_{k=0}.$$ (29.103)

The tracer flux F^z_k=0 must be calculated from a boundary condition which equates this flux with the total tracer flux Q_Tthrough the surface. The flux Q_T generally has a contribution from parameterized turbulence as well as a tracer flux with fresh water,

 

$$Q_{T} = Q_T^{turb} - q_w \, T_{w},$$ (29.104)

where T_w is the tracer concentration in the fresh water. Although T_w and T_k=0 may be of the same order of magnitude, the terms $q_w \, T_{w}$ and $q_w \, T_{k=0}$ stand for different physical processes. The term $q_w \, T_{w}$ represents the amount of tracer passing through the air sea interface with fresh water, whereas $q_w \, T_{k=0}$ represents the advection of tracer in the ocean relative to the sea surface. Note that for the special case of salt, the total surface salt flux Q_s is well approximated to be zero for climate modeling. More will be said about salt below.

In general, specification of the surface tracer flux involves details of how the air-sea interface is modeled. A simple example of a ``closure'' for the turbulence term is a restoring condition $Q_T^{turb} = \gamma \, h^{t} \, (T_{1} - T^{*})$, where $\gamma$ is an inverse damping time and T₁ is the time lagged surface tracer concentration. For the advection term, a simple choice is to set the tracer concentration T_w in the fresh water equal to the surface cell concentration T₁, now without time lag as it represents an advective contribution. Other forms are generally appropriate when employing ocean models with more realistic surface boundary conditions, such as for coupled climate simulations.